The present invention concerns the procedure for analyzing a FT-IR-spectrum of an unknown mixture of gases, about which we do not know the constituent gases, not to mention their partial pressures. Instead we know a large set of library spectra of pure molecular gases, measured in known pressures and with the same interferometer as the unknown sample.
1. Field of the Invention
In FT-IR-spectroscopy in multicomponent analysis we have the IR-spectrum of an unknonw mixture of gases, about which we co not inow the constituent gases, not to mention their partial pressures. Instead we know a large set f library spectra of pure molecular gases, measured in known pressures and wih the same interferometer as the unknown sample. By using these pure spectra we ought to calculate the patial pressures of the pure gases in the mixture, with error limits. The errors in the obtained values arise from the measurement noise in the spcectra. Because the calculation of such partial pressures that best explain the mixture spectrum is a relatively simple task, we shall consider it only briefly and concentrate on the calculation of their error limits. We chall also consider the optimal choice of the resolution providing as small error limits as possible and the application of the non-negativity constraint for the partial pressures. In addition to gas spectra, all the calculations apply to spectra of non-interacting liquids as well.
2. Description of the Prior Art
Let us denote by s the measured mixture spectrum to be analyzed and let {K.sup.j .vertline.j=1, . . . ,m} be the set of library spectra with which we are trying to explain s. We assume that all the spectra are linearized, i.e. that on the y-axis there is always the negative logarithm of the transmittance (except possible some constant coefficient). This means that we are using the absorbance scale. Then, according to Beer's law, s is some linear combination of the spectra K.sup.j, or ##EQU1## Our task is to determine such coefficients x.sub.j by which s is explained as well as possible. After that the partial pressures of the pure gases in the mixture ar obtained by multiplying the measuring pressures of the corresponding library spectra by their coefficients in the linear combination.
Because all the spectra are only known at equally spaced tabelling points .upsilon..sub.1, . . . ,.upsilon..sub.n, s and k.sup.j can be treeated as vectors, and we can state Eq. 1 in vector form as ##EQU2## This equation can be further simplified by collecting all K.sup.j -vectors together into a matrix K, which gives us the matrix equation EQU Kx=s, (3)
where ##EQU3## and x is the coefficient (column) vector containing the coefficients x.sub.j of the pure spectra.